ABSTRACT
Let E be a 2-uniformly convex and uniformly smooth real Banach space with dual space E. Let
A : C ! E be a monotone and Lipschitz continuous mapping and U : C ! C be relatively nonexpansive.
An algorithm for approximating the common elements of the set of fixed points of a
relatively nonexpansive map U and the set of solutions of a variational inequality problem for the
monotone and Lipschitz continuous map A in E is constructed and proved to converge strongly.
TABLE OF CONTENTS
Certification i
Approval ii
Abstract iii
Acknowledgement iv
Dedication v
1 Introduction 1
1.1 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Variational inequality problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Fixed Point Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Review 4
2.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Nonexpansive Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Theory and Methods 9
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Metric Projection Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Calculating the projection onto a closed convex set in Hilbert spaces . . . . . 19
4 Main Result 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
vi
5 Application 29
5.1 Strong Convergence Theorem for a Countable Family of Relatively Nonexpansive
Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Conclusion 31
Bibliography 32
CHAPTER ONE
1.1 Background of study
The notion of monotone operators was introduced by Zarantonello [Zarantonello, 1960], Minty
[Minty, 1962] and Ka˘ curovskii [Ka˘ curovskii, 1960]. Monotonicity conditions in the context of
variational methods for nonlinear operator equations were also used by Vainberg and Ka˘ curovskii
[Vainberg et al., 1959].
A map A : D(A) H !H is monotone if
hAx Ay;x yi 0 8x;y 2 H:
Consider the problem of finding the equilibrium states of the system described by
du
dt
+ Au = 0; (1.1)
where A is a monotone-type mapping on a real Hilbert space. This equation describes the evolution
of many physical phenomena which generate energy over time. It is known that many
physically significant problems in different areas of research can be transformed into an equation
of the form
Au = 0: (1.2)
At equilibrium state, equation (1.1) reduces to equation (1.2) whose solutions, in this case, correspond
to the equilibrium state of the system described by equation (1.1). Such equilibrium points
are very desirable in many applications, for example, economics, ecology, physics and so on.
1
1.2 Variational inequality problem
Let C be a nonempty, closed and convex subset of a real normed space E with dual space E.
Let A : C E ! E be a nonlinear operator. The classical variational inequality problem is the
following: find x 2 C such that
hAx;y xi 0 8 y 2 C: (1.3)
The set of solutions of inequality (1.3) is denoted by V I(C; A). The variational inequality problem
is connected with convex minimization, fixed point problem, zero of nonlinear operator and
so on.
Variational inequality has been shown to be an important mathematical model in the study of
many real problems, in particular equilibrium problems. It provides us with a tool for formulating
and qualitatively analyzing the equilibrium problems in terms of existence and uniqueness of
solutions, stability, and sensitivity analysis, and provides us with algorithms for computational
purposes.
For example, in optimization, we consider f : [a;b]!R differentiable. It is well known that such
f has a minimizer, say x 2 [a;b]. We have the following cases:
1. x = a)f 0(x) (x x) 0 8 x 2 [a;b]
2. x = b)f 0(x)(x x) 0 8 x 2 [a;b]
3. x 2 (a;b))f 0(x)(x x) = 0 8 x 2 [a;b]
Thus, setting C = [a;b], A = f 0 we have
x is a minimizer)hAx;x xi 0 8 x 2 C:
In general, in Euclidean n-dimensional Rn, the variational inequality (1.3) becomes (yx)>Ax
0 8 y 2 C: This is equivalent to y>Ax x>Ax 8 y 2 C: Thus, x is a solution to the minimization
problem 8>>>><>>>>:
miny>Ax;
y 2 C;
i.e.
x 2 V I(C; A),x solves
8>>>><>>>>:
miny>Ax;
y 2 C:
2
1.3 Fixed Point Problem
In 1922, Banach [Banach, 1922] published his fixed point theorem known as Banach’s Contraction
Mapping Principle using the concept of Lipschitz mapping. A fixed point of an operator T is a
solution of the equation x = T x. The set of fixed points of T is denoted by F(T ). T is called a
contraction if there exists a fixed L < 1 such that
kT x T yk Lkx yk for all x;y 2 E: (1.4)
A contraction mapping is also known as a Banach contraction. If inequality (1.4) holds for L = 1,
then T is called nonexpansive and if inequality (1.4) holds for fixed L < 1, then T is called
Lipschitz continuous. Clearly, for the mapping T , the following obvious implications hold:
Contraction =) Nonexapansive =) Lipschitz continuous
The concept of fixed points makes sense only when the map T maps the space into itself, but this
concept does not make sense when T maps the space into its dual.
Our main focus in this thesis is to construct an iterative algorithm that converges strongly to a solution
of the set of fixed point problems of a relatively nonexpansive mapping and the set of variational
inequality problems for monotone and Lipschitz continuous mapping on a 2-uniformly
convex and uniformly smooth real Banach space.
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